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Does cocoa contain caffeine?

Yes, in very small quantities.

Cocoa beans

Cocoa beans undergo a process of drying and fermentation to develop the flavour and colour. The amount of caffeine in the beans varies with the type of beans and the degree of fermentation.

In an analysis of cocoa from various sources the following percentages of caffeine were found:

Cocoa powder

Cocoa powder contains caffeine. Depending on the degree of fermentation and the type of cocoa beans, the caffeine content will vary from 0.1-0.5%. In cocoa powders made from well fermented African cocoa beans the caffeine proportion is in general very low: 0.1% or less.

Chocolate

Levels of caffeine range from 2 to 23 milligrams in a typical 40 gram bar of milk chocolate and from 2 to 7 milligrams in a 227 gram glass of chocolate milk; both are the equivalent of that found in cups of decaffeinated coffee. A 28 gram piece of bittersweet (dark) chocolate contains from 5 to 35 milligrams of caffeine, still well under the approximately 75 milligrams of caffeine found in a 227 gram cup of instant coffee or the 140 milligrams found in the same amount of brewed coffee.

Cocoa butter

A study by Kiefer & Martin to determine the amounts of caffeine and theobromine in chocolate products found only small amounts of theobromine (0.008%) and caffeine (0.038%) in cocoa butter.

References:
The cocoa manual. Cacao de Zaan, 1993, p 91-92)
Chocolate and health. A scientific overview for the health professional. Chocolate Manufacturers Association of the USA, 1997